Financial mathematics: percentages, GST and simple interest
Percentage mark-ups and discounts, adding and removing 15% GST, and calculating simple interest
About three lessons of 45 to 60 minutes
How much of $115 is actually GST?
A price tag reads '$115, GST included'. It's tempting to find the GST by taking 15% of $115, which gives $17.25. But that overstates it: the $115 already INCLUDES the tax, so taking 15% of the GST-inclusive price double-counts the effect the tax itself had on the total.
The $115 is actually 115% of the price before GST (100% for the item, plus 15% GST on top), so the price before GST is $115 / 1.15 = $100 exactly, and the real GST component is $115 - $100 = $15, not $17.25. This unit is about telling the two directions apart, ADDING GST to a price you already know, and pulling GST back OUT of a total that already includes it, plus the closely related skills of percentage mark-ups and discounts, and simple interest, which all use the same 'percent of an amount' idea underneath.
- A $200 job plus 15% GSTinvoiced at $230, the extra $30 is the GST
- A $115 price tag marked 'GST included'the price before GST is $100, not $115 x 0.85
- A $50 item marked up 20% for resalesells for $60, the shop's mark-up is $10
- $500 invested at 4% p.a. simple interestearns $20 every year it stays invested
What students will be able to do
Students will apply a percentage mark-up or discount to a price and work backwards from the new price to the original, find a total price including 15% GST from a GST-exclusive price, find the GST component of a GST-inclusive total without simply taking 15% of it, and calculate simple interest (and rearrange the relationship to find a missing rate or time) using simple interest = principal x rate x time / 100.
- I can apply a percentage mark-up or discount to a price to find the new price.
- I can work backwards from a price after a mark-up or discount to find the original price.
- I can find the total price including 15% GST, given the price before GST.
- I can find the GST component of a GST-inclusive total, without simply taking 15% of the inclusive price.
- I can calculate simple interest using I = P x R x T / 100, and find the total amount in an account.
- I can rearrange the simple interest relationship to find a missing interest rate or time.
Standards this unit teaches
- Phase 4 (Years 9-10): Number, Financial mathematicsNew Zealand Curriculum (NZC), Mathematics and StatisticsFinancial mathematics
Paraphrased (see the licence note in this file) from the Ministry of Education's Tāhūrangi curriculum site, "NZC - Mathematics and Statistics Phase 4 (Years 9-10)" (official policy for all English-medium state and state-integrated schools from 1 January 2026), Number strand, "Financial mathematics" section: during Year 9, students apply percentage mark-ups and discounts, and calculate simple interest and GST on dollar amounts (the source's own example finds 15% GST on $432); during Year 10 this extends to converting NZ dollars to other currencies and to compound interest, calculated month by month. New Zealand's GST rate is 15%. Source: https://newzealandcurriculum.tahurangi.education.govt.nz/nzc---mathematics-and-statistics-phase-4-years-9-10/5637291579.p (verified live 2026-07-14).
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Grade 7 ratios, proportions and percent teaching unitfinding a percentage of an amount, the core skill every calculation in this unit builds on
- Grade 6 percentages teaching unitconverting between fractions, decimals and percentages
- Percentage in the glossarya quick refresher on what a percentage means
- Ratio in the glossarythe same part-whole thinking underlies a GST-inclusive total split into its parts
Words to teach and display
- Mark-up
- the percentage a seller adds to what they paid for an item, to set its selling price
- Discount
- the percentage taken OFF a price, usually for a sale or promotion
- GST
- Goods and Services Tax, a New Zealand tax of 15% added to the price of most goods and services
- GST-inclusive
- a price that already has GST added in, the way almost all New Zealand price tags are shown
- Simple interest
- interest calculated only on the original amount invested or borrowed (the principal), not on interest already earned
- Principal
- the original amount of money invested or borrowed, before any interest is added
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Percentage mark-ups and discounts
ConcreteA shop buys a $50 item and wants to make a profit, so it adds a mark-up: a percentage of the cost price, added on top. A 20% mark-up on $50 means finding 20% of $50 (which is $10) and adding it: $50 + $10 = $60. A discount works the same way in reverse, a percentage is found and then SUBTRACTED.
Both mark-ups and discounts start with the same first step: find the percentage of the original amount. Whether that amount is then added (mark-up) or subtracted (discount) is the only difference between the two calculations.
Working backwards is trickier. If a jacket now costs $68 after a 15% discount, you cannot just add 15% of $68 back on, that would be 15% of the WRONG amount (the already-discounted price). Instead, recognise that $68 is 100% - 15% = 85% of the original price, so the original price is $68 / 85 x 100 = $80.
A jacket costs $80 before a 15% discount. What is the sale price? Then, if you only knew the sale price was $68, how would you find the original $80 back?
- Forward: 15% of $80 = $80 x 15/100 = $12. Sale price = $80 - $12 = $68.
- Backward: $68 is 100% - 15% = 85% of the original price (not 100% of it).
- Original price = $68 / 85 x 100 = $80.
Answer: The sale price is $68, and working backwards from $68 correctly recovers the original $80.
- Why can't you find the original price after a discount by just adding the same percentage back on to the new price?
- A $40 item gets a 25% mark-up. What single number (mark-up amount) do you calculate first?
2. GST: adding and removing 15%
PictorialGST (Goods and Services Tax) is a New Zealand tax of 15%, added to the price of most goods and services. Adding GST to a price you already know (GST-exclusive) works exactly like a mark-up: find 15% of it, then add. A $200 job before GST becomes $200 + $30 = $230 including GST.
The tricky direction is the reverse: almost every price tag in New Zealand is already GST-INCLUSIVE, so you often need to find how much of a total is GST, or what the price was before GST, starting from the inclusive total. The inclusive total is 115% of the original (100% for the item plus 15% GST), so dividing by 115 and multiplying by 100 (equivalently, dividing by 1.15) recovers the price before GST.
The classic mistake is taking 15% of the GST-INCLUSIVE total to find the GST component. $115 x 15% = $17.25, but the real GST hidden inside $115 is only $15 (because the $100 it was added to is smaller than $115). Always find the pre-GST price FIRST, then subtract it from the total to get the true GST amount.
The price before GST for a job is $200. Find the total including GST. Then, given only the $230 GST-inclusive total, find the GST component.
- Forward: GST = $200 x 15/100 = $30. Total including GST = $200 + $30 = $230.
- Backward: $230 is 115% of the price before GST, so the price before GST = $230 / 115 x 100 = $200.
- The real GST component = $230 - $200 = $30 (NOT 15% of $230, which would give $34.50).
Answer: The GST-inclusive total is $230, and working backwards from $230 correctly recovers a $200 pre-GST price and a $30 GST component.
- Why does taking 15% of a GST-inclusive total overstate the real GST amount?
- What single number do you divide a GST-inclusive total by to find the price before GST?
3. Simple interest
AbstractSimple interest is interest calculated only on the original amount invested (the principal), every single year, not on interest already earned. The formula is simple interest = principal x rate x time / 100, often written I = PRT / 100.
$500 invested at 4% per year (p.a.) earns $500 x 4/100 = $20 every single year, because simple interest is always calculated on the ORIGINAL $500, never on a growing balance. After 3 years the total interest is $20 x 3 = $60 (matching I = PRT/100 = 500 x 4 x 3 / 100 = $60 directly), and the total amount in the account is the principal plus the interest: $500 + $60 = $560.
The same formula rearranges to find a missing rate or time, when the interest, principal and the other two values are known. Finding the rate: rate = interest x 100 / (principal x time). Finding the time: time = interest x 100 / (principal x rate). Both rearrangements simply undo the multiplication in the original formula.
$500 is invested at 4% p.a. simple interest for 3 years. Find the interest earned and the total amount. Then, given only that $300 invested for 3 years earned $36 interest, find the interest rate that was used.
- Interest: I = P x R x T / 100 = $500 x 4 x 3 / 100 = $60. Total amount = $500 + $60 = $560.
- Rate: rearranging I = PRT/100 gives R = I x 100 / (P x T) = $36 x 100 / ($300 x 3) = 3600 / 900 = 4%.
Answer: $500 at 4% p.a. for 3 years earns $60 interest (total $560), and $300 earning $36 over 3 years used a rate of 4%.
- Why does $500 at 4% p.a. simple interest earn exactly the same $20 in year 1, year 2 and year 3?
- Which two values in I = PRT/100 do you multiply together first, before dividing, to find a missing rate?
Common misconceptions and how to address them
MisconceptionTo find the GST inside a GST-inclusive total, just take 15% of that total.
Why it happens: Students apply the GST percentage to the wrong base: the inclusive total is already 115% of the original price, not 100% of it, so 15% of the total is 15% of an amount that is itself already too big.
How to address it: Always find the price BEFORE GST first, by dividing the inclusive total by 1.15 (or by 115 then multiplying by 100). Then subtract that from the total to get the real GST amount. $115 x 15% = $17.25 is wrong; the real GST inside $115 is $15.
MisconceptionTo find the original price before a discount, add the discount percentage back on to the new (discounted) price.
Why it happens: Students treat the discount percentage as if it applies equally in both directions, forgetting the discounted price is a SMALLER base than the original, so the same percentage of it is a smaller dollar amount.
How to address it: The discounted price is (100 - discount%) of the ORIGINAL, not the other way round. Divide the discounted price by (100 - discount%), then multiply by 100, to correctly recover the original price.
MisconceptionSimple interest grows the amount it is calculated on each year, the way a savings balance actually grows.
Why it happens: Students confuse simple interest with compound interest (next year's topic): they assume the interest earned in year 1 becomes part of the amount interest is calculated on in year 2.
How to address it: Simple interest is always calculated on the ORIGINAL principal, every year, so the dollar amount of interest earned is identical each year. $500 at 4% p.a. earns exactly $20 in year 1, year 2, and every following year, it never compounds.
MisconceptionA mark-up and a discount of the same percentage cancel out (e.g. a 20% mark-up followed by a 20% discount returns the original price).
Why it happens: Students assume percentages of DIFFERENT amounts are directly comparable, because the percentage number is the same.
How to address it: A 20% mark-up on $50 gives $60 (20% of $50). A 20% discount on that NEW $60 removes 20% of $60 = $12, giving $48, not back to $50. The two 20%s are percentages of different base amounts, so they do not cancel.
Guided practice (with answers)
1. A $40 item has a 25% mark-up added. Find the new price.
Answer: $50, because 25% of $40 = $10, and $40 + $10 = $50.
2. A $90 item is discounted by 10%. Find the sale price.
Answer: $81, because 10% of $90 = $9, and $90 - $9 = $81.
3. After a 20% discount, an item costs $56. What was the original price?
Answer: $70, because $56 is 100% - 20% = 80% of the original: $56 / 80 x 100 = $70.
4. Find the total price including 15% GST for a $60 service (before GST).
Answer: $69, because GST = $60 x 15/100 = $9, and $60 + $9 = $69.
5. A price of $138 includes 15% GST. Find the price before GST.
Answer: $120, because $138 is 115% of the price before GST: $138 / 115 x 100 = $120.
6. Find the simple interest on $800 invested at 5% p.a. for 2 years.
Answer: $80, because I = P x R x T / 100 = $800 x 5 x 2 / 100 = $80.
Independent practice worksheets
Practise percentage mark-ups and discounts, GST in both directions, and simple interest, with computed, never-wrong answer keys.
Differentiation
- For discounts and mark-ups, always write the percentage as a fraction out of 100 first (e.g. 15% = 15/100) before multiplying, rather than trying to convert to a decimal mentally.
- For GST-inclusive questions, give the 115% (or 85% for a discount) explicitly at first, so the only step left is the division.
- Use $100 as the starting principal for early simple-interest questions, so the interest for 1 year is simply the rate itself in dollars (e.g. 4% of $100 is $4).
- Provide a written checklist for the GST-reverse question: 1) divide by 1.15 to get the price before GST, 2) subtract that from the total to get the GST amount.
- Investigate: does a 10% mark-up followed by a 10% discount return the original price? Explain why or why not using two different bases.
- Pose a two-step problem: an item is marked up 20%, then later discounted 20% off the NEW price. Find the final price as a percentage of the true original.
- Compare the mathematical structure of adding 15% GST with applying a 15% mark-up, and ask students to explain why the 'divide by 1.15' method for reversing either percentage increase is identical.
- Preview compound interest by asking: if the year 1 interest were reinvested and ALSO earned interest in year 2, would the year 2 interest be more, less, or the same as year 1's? (More, this is the difference from simple interest, and is next year's topic.)
Assessment: exit ticket
A three-question exit ticket sampling a mark-up/discount, a GST-inclusive reversal, and simple interest.
1. A $70 item is discounted by 20%. Find the sale price.
Answer: $56, because 20% of $70 = $14, and $70 - $14 = $56.
2. A price of $92 includes 15% GST. Find the price before GST.
Answer: $80, because $92 is 115% of the price before GST: $92 / 115 x 100 = $80.
3. Find the simple interest on $600 invested at 5% p.a. for 4 years.
Answer: $120, because I = P x R x T / 100 = $600 x 5 x 4 / 100 = $120.
Teacher notes and timings
- Rough timing across three lessons: Lesson 1 percentage mark-ups and discounts, including working backwards (section 1), Lesson 2 GST in both directions, with the GST-inclusive misconception explicitly taught (section 2), Lesson 3 simple interest and its rearrangements, plus the exit ticket (section 3).
- This is the site's first teaching unit anchored to the New Zealand Curriculum (NZC). See the licence note above this unit's definition in lib/teachingUnits_secondary.ts: unlike the UK's Open Government Licence v3.0 unit above (ukYear7RatioAndProportion, quoted verbatim), the NZC's Tāhūrangi content is licensed CC BY-NC 4.0 (NonCommercial), so this unit paraphrases the curriculum rather than quoting it, with the source URL cited for attribution.
- Scope note: this unit deliberately covers Year 9's financial mathematics practices (percentage mark-ups/discounts, simple interest, GST) and leaves Year 10's continuation (currency conversion, compound interest calculated month by month) for a later unit, matching how the source page itself splits Year 9 from Year 10.
- New Zealand's GST rate is 15%, not Australia's 10% (see lib/content_ausecondarymath.ts's financial mathematics worksheet, which correctly uses 10% for an Australian audience). Keep the two rates separate when comparing notes across the AU and NZ content.
- Language to keep repeating: a GST-inclusive (or post-discount, or post-mark-up) amount is a DIFFERENT percentage base than the original, so you cannot apply the same percentage figure directly to it in the reverse direction.
- Use Student view to project this lesson. Print saves the full teacher unit, including answers and teacher notes; use the linked independent-practice worksheets for student handouts.